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Integral with x2−a2\sqrt{x^2-a^2}

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∫1x2−a2 dx=x∣x∣arcosh⁡∣x∣a+C=ln⁡∣x+x2−a2∣+C\int{\frac{1}{\sqrt{x^2-a^2}}}\ \mathrm{d}x=\frac{x}{\left|x\right|}\operatorname{arcosh} {\frac{\left|x \right|}{a}}+C=\ln{\left|x+\sqrt{x^2-a^2}\right|}+C
Derivation

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∫1(x2−a2)3 dx=−xa2x2−a2+C\int{\frac{1}{\sqrt{(x^2-a^2)^3}}}\ \mathrm{d}x=-\frac{x}{a^2\sqrt{x^2-a^2}}+C
Derivation

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∫xx2−a2 dx=x2−a2+C\int{\frac{x}{\sqrt{x^2-a^2}}}\ \mathrm{d}x=\sqrt{x^2-a^2}+C
Derivation

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∫x(x2−a2)3 dx=−1x2−a2+C\int{\frac{x}{\sqrt{(x^2-a^2)^3}}}\ \mathrm{d}x=-\frac{1}{\sqrt{x^2-a^2}}+C
Derivation

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∫x2x2−a2 dx=x2x2−a2+a22ln⁡∣x+x2−a2∣+C\int{\frac{x^2}{\sqrt{x^2-a^2}}}\ \mathrm{d}x=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\ln{\left|x+\sqrt{x^2-a^2} \right|}+C
Derivation

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∫x2(x2−a2)3 dx=−xx2−a2+ln⁡∣x+x2−a2∣+C\int{\frac{x^2}{\sqrt{(x^2-a^2)^3}}}\ \mathrm{d}x=-\frac{x}{\sqrt{x^2-a^2}}+\ln{\left|x+\sqrt{x^2-a^2} \right|}+C
Derivation

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∫1xx2−a2 dx=1aarccos⁡a∣x∣+C\int{\frac{1}{x\sqrt{x^2-a^2}}}\ \mathrm{d}x=\frac{1}{a}\arccos{\frac{a}{\left|x\right|}}+C
Derivation

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∫1x2x2−a2 dx=x2−a2a2x+C\int{\frac{1}{x^2\sqrt{x^2-a^2}}}\ \mathrm{d}x=\frac{\sqrt{x^2-a^2}}{a^2x}+C
Derivation

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∫x2−a2 dx=x2x2−a2−a22ln⁡∣x+x2−a2∣+C\int{\sqrt{x^2-a^2}}\ \mathrm{d}x=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln{\left|x+\sqrt{x^2-a^2} \right|}+C
Derivation

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∫(x2−a2)3 dx=x8(2x2−5a2)x2−a2+38a4ln⁡∣x+x2−a2∣+C\int{\sqrt{(x^2-a^2)^3}}\ \mathrm{d}x=\frac{x}{8}(2x^2-5a^2)\sqrt{x^2-a^2}+\frac{3}{8}a^4\ln{\left|x+\sqrt{x^2-a^2} \right|}+C
Derivation

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∫xx2−a2 dx=13(x2−a2)3+C\int{x\sqrt{x^2-a^2}}\ \mathrm{d}x=\frac{1}{3}\sqrt{(x^2-a^2)^3}+C
Derivation

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∫x2x2−a2 dx=x8(2x2−a2)x2−a2−a48ln⁡∣x+x2−a2∣+C\int{x^2\sqrt{x^2-a^2}}\ \mathrm{d}x=\frac{x}{8}(2x^2-a^2)\sqrt{x^2-a^2}-\frac{a^4}{8}\ln{\left|x+\sqrt{x^2-a^2} \right|}+C
Derivation

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∫x2−a2x dx=x2−a2−aarccos⁡a∣x∣+C\int{\frac{\sqrt{x^2-a^2}}{x}}\ \mathrm{d}x=\sqrt{x^2-a^2}-a\arccos{\frac{a}{\left|x\right|}}+C
Derivation

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∫x2−a2x2 dx=−x2−a2x+ln⁡∣x+x2−a2∣+C\int{\frac{\sqrt{x^2-a^2}}{x^2}}\ \mathrm{d}x=-\frac{\sqrt{x^2-a^2}}{x}+\ln{\left|x+\sqrt{x^2-a^2} \right|}+C
Derivation